At no point should 'mass' ever be treated like 'inertia'. Newton's law says f=ma. This 'law' is supposed to be a 'scientific' one and so it must be possible to test it. We must be able to push a stone of mass m, using a force of f and see if it accelerates at a. So we must have a way of knowing the value of 'm' before we accelerate the object, so we may test if it indeed accelerates as specified by the f=ma so called 'law'. Given that this is the case, how comes physicist measure mass, for instance using 'mass spectrometer', by pushing an object with a known force, f , watching how it accelerates and then shouts that they have 'measured the mass' and found it to be given by m=f/a? Whenever you see a team of physicists measuring 'mass' by pushing objects and watching the movie of their acceleration, know at once that you are dealing with a gang of idiots!

If an object or a particle has not grown fatter due to addition of more particles, nor have we watched it get compressed before other stuffs are added to it in order to return it to its original size, there is no meaningful way of saying that 'its mass has increased'. If, nevertheless, we see that its inertia has changed, then we don't say 'its mass has changed'. Rather, we simply say that Newton's second law is not scientifically correct because inertia of an object can alter without any observed alteration of its matter content. Therefore when a physicist measures atomic masses using mass spectrometer and then say 'there is a mass deficit' (converted to energy?) he is behaving like a lunatic! An object can only alter its mass via addition/ removal of more matter to the object in question. This addition/removal must be observable, just like when you are adding more floor to make an heavier bread! You can never infer addition of more matter by pushing the object and the watching how it moves. That way you can only observe its altered BEHAVIOUR and never, its altered INTRINSIC PROPERTY or amount of anything that constitutes the object itself, as opposed to its behaviour.

We are going to close examine that inertia is a thoroughly distinct property from 'mass'. I am going to emphasize the falsifiability of Newton's law, f=ma by showing how an object can alter its inertia without altering its mass. I will show you that this is the case in Electromagnetic Waves (EM waves). But first, we will close examine the usual, 'mechanical' waves.

If you want to pluck a ruler, or a similar object, to make it vibrate, you begine by pinning one of its end on a table, or on a similar object. So we have the ruler's free edge and its pinned edge. Before the ruler is plucked, it is in a rest state, which is straightened and horizontal. To pluck the ruler, you press down its free edge. So originally, before you release the ruler to allow it to vibrate it is curved downwards so that its free edge is taken to the furthest point away from where it was at rest state. Being the furthest means that it will take the longest time for it to move back up to it's place when it is at rest.

Now instead of a vibrating ruler, think of a more flaccid but springy 'ruler' or a similar object. Imagine holding one end so that there is the held edge and then the free edge. Initially, the flaccid object is lieing horizontaly (though not perfectly so, due to gravity). Now if you move the held edge suddenly upwards, the object will now breifly assume a shape bent downwards so that it now resembles the pressed down 'ruler' we saw above. This is because of the inertia of the points along the ruler, together with the ruler not being riggid enough to take all the points at once. But if you suddenly stop moving the object upwards, the object will now re-straighten itself at the more elevated position, and the inertia together with some degree of the springiness, may make the object vibrate briefly just like the plucked ruler. When you stopped moving it 'upwards', the held edge reached its rest state at exactly that instance, but the free edge took some times to reach its rest state. This is because the bending 'down' caused by the up movement of the held edge combined with the inertia of the points along the 'ruler', resulted in the free edge getting left at a point farthest from its rest point, just like in the bent ruler.

Now instead of moving the held edge 'upwards' and then suddenly stopping it, move it upwards up to somewhere and then suddenly move it back 'downwards'. The inertia of the points along the ruler will make the ponts farther away from the held edge keep moving 'upwards' even after you have begun to move the held edge 'downwards'. However, remember also that the points even further away from the held edge were left more 'down' when you were moving the held edge 'upwards', with the farthest edge being the most 'downwards'. All this results in a ruler that assumes a shape bent 'upwards' upto some point, before it now gets bent 'downwards'(an hill-like shape). So it has some 'highest' point somewhere along the ruler, like the summit of an hill.

Think of the ruler now as two rulers, call it ruler A and B, joined together at the 'summit' point. So ruler A is bent 'upward' upto the summit and then there, it gets joined to ruler B, which is bent 'downwards' all the way upto the free edge. So 'ruler B', is farther away from the held edge than 'ruler A' is. Note that these 'two rulers' are now in a state similar to the rulers that are about to be plucked rulers that are about to be released to vibrate. They are under tension due to the springiness. The 'ruler A' tries to bend back 'downwards' while the 'ruler B', due to inertia, tries to keep bending back 'upwards'. Since 'ruler B' is farther away from the held edge than the 'ruler A' is, the summit point gets moved downwards by the bending of ruler A while the point just further away from the held edge than the summit is gets moved upwards by the bending of ruler B, and it becomes the new summit. So the summit moves further away from the held edge. This is the wave moving away from the held edge.

Note that if you begun to move the held edge upwards and then suddenly stop at some time t0, then the farthest, free edge will move up to the 'rest' state at some latter time, t1, so that t1-t0=the half of the period of the full oscillation cycle. This is to say the natural frequency,f, of the vibrating ruler is 1/2 (t1-t0). Then if L=the length of the 'ruler', then L/(t0-t1)=2Lf, must be the speed of the wave.

The foregoing detailed description is meant to illustrate the two factors that are interplaying to cause wave propagation: the inertia and the 'springiness' of the medium. It also shows that if someone talks of 'waves' in a completely empty space, then that person did not do the homework of understanding the simpler acoustic waves, let alone the EM waves! Often he is actually confusing 'knowing how to describe motion mathematically' with 'understanding how the phenomenon is actually happening'. In the mathematical description, we cannot even differentiate a 'translation of a wavy object' with the 'waving of a straight object, so that only the waviness, not the object itself, gets translated'. They are both described the same way using the same stupid equation!

The perhaps surprising fact is that to understand the EM waves, we don't need any different insight from the one above. We only need to identify what it is that is behaving like 'springiness' and what it is that is behaving like inertia in an apparently empty space. Then the explanation will proceed exactly like in the so called 'mechanical wave' like we saw above. You will note that the property that is behaving like 'inertia', unlike in the 'mechanical wave', is realy not the 'mass', and is certainly not, 'the quantity of matter'. You will also note more clearly that the 'm' in the equation e=mc^2 refers to this other 'inertia', and never to 'quantity of matter'. Then also, you will see what 'speed of light' (EM waves) got to do with it, and perhaps thinking of the inertia in the 'mechanical wave' has already hinted it to you!

The speed of the wave along a string (related to the natural frequency of the string like we saw above) is given by:

v=sqrt (T/ρ),

where ρ is 'inertia density' ('mass' per length) of the string and T is its tension ('springiness') . This 'inertia' have come to be equated with 'mass' due to imprudent taking of the correctness of Newton's laws for granted! So we can rewrite:

T= ρv^2 or
Tdx=mv^2

Where dx is 'small distance interval' between some two points along the string and m is the 'amount of inertia' (erroneously equated with 'mass') sitting within the small 'dx' portion of the string. So ρ=m/dx. We then see that Tdx (tension times small distance) is the amount of potential energy existing in the 'dx' region. It is the energy stored by the tension streatching the somewhat springy string by the small 'dx' amount. So you see how potential energy in a region is equal to the 'mass' at that region times the speed of a wave squared. The reasoning is the same for the speed of light e=mc^2, having in mind that light is an EM wave. Then by the way note that we don't need any relativity theory to see such!

It is important to understand that an accelerated, charged particle, behaves like it has some inertia, even if it were 'massless' in the Newtonian sense. In other words charge plus acceleration is an 'inertia' on its own, without any 'mass'! Charge also has some 'springiness' in the sense that if you try to pull a charge off another opposite charge, it gets pulled back. We can use this to figure out EM waves if we think of 'empty space' rather as to be composed of charges of opposite sign. (This also explains the so called 'vacuum polarization').

The 'charge inertia' is caused by magnetism. A moving charge creates magnetism around itself. This magnetism, in turn, creats an electric current that moves in the direction opposite to the moving charge. This induced electric current opposes the original motion of the very charge. Therefore the charge has some resistance to change of its velocity that is not related in anyway to its mass! The 'induced current' is capable of affecting the velocity of the charge because electric current is actually a 'flow of charges'.

A single charged particle that is moving can be seen as an electric current. Infact 'electric current' is defined as 'the amount of charge crossing a region per given time'. To get it more clearly, think of an object of charge q, and of length x crossing some region R. Its leading edge is at R at some time , and its trailing edge is at the same R at some later time t1. So it is etering R at t0 and exiting it at t1. So the speed of the object, v, is given by v=x/(t1-t0). But the amount of charge crossing R at the time interval will be q, and so the electric current, I, will be given by: I= q/(t1-t0)=qv/x=qvA'/A'x=ρA'v, where A' is the area of the region and ρ is the 'charge density', noting that A'x is the volume in which the charge is sitting. It is in this way 'charge velocity' is related to 'electric current'.

Now the acceleration, a, of the charged particle, given by a=dv/dt,(read 'change in velocity/change in time) will be given by a=(x/q)*(dI/dt). That is given the equation I=qv/x, you note that the change in velocity,v, per given time leads to a coresponding change in current per the given time. So rate of change in electric current is propotional to the acceleration of charge. We will next see that the laws of electromagnetism introduces a kind of inertia (resistance to the acceleration) to the charge that is not, in any way, related to its mass but instead is only related to its charge and magnetism. The following image puts this more clearly:

To begine noticing the inertia, consider voltage across some conductor. When current flows through the conductor, it creates magnetism. The magnetism itself, when it changes, creates an electric current, through the same conductor. This induced current, when combined with the iriginal current causes of the 'inertia'. The voltage across the conductor is the source of the force that drives the charges through the conductor. We will use a law called 'Biot-Sarvat law' to describe the magnetic field caused by the electric current flowing around a loop of 'wire' (it will approximate a vortex ring in space), and then use Faraday's law to discribe how the changing magnetism induces voltage across the very same conductor. When the Faraday's law is combined with the Biot-sarvat law, we arrive at an equation that looks like Newton's Second law of motion albeit with a ficticious 'mass' that is not even an intrinsic property of the accelerated charge.

Voltage is related to force because it is defined as 'work done per unit charge'. If you pull a charge, for some distance, away from another charge of opposite sign, you perform some 'work' on it given by:w=F*distance, when we consider a distance around a circular loop, we have w=F*2π r, where F is 'force'. So voltage V will be given by: V=F*2πr/q. The following is the calculation:

Note the following:

1.the first line is the Biot-Savart  Law for magnetism within a single loop of a circular conductor. μ0 is magnetic permeability and B is the magnetic field and r is the radius of the circular conductor.

2.In the second line, I take the derivative with respect to time on both sides of the Biot-Sarvat law, which allows me to relate the change in magnetic field with time to the same change in Faraday's law, which I introduce it in the third line

3.In the fourth line, I combine the Biot-Savart law with Faraday's law by subtituting for dB/dt the A appearing in Faraday's Law is the area of the circle whose radius is r and the A' is crossection area of the conductor, ie the region R that we saw earlier.

4.In the 5th line, I use the relationship between voltage (V) and work and then force (F) to rewrite the combination of Biot Sarvat Law with Faraday's law. I also subtitute A=πr^2. We also notice that the equation is now similar to Newton's Second law F=mdv/dt. This allows me to recognize the ficticious 'mass' (m') that I point it out in the last line.

Also take note of the following:

1.)The ficticious 'mass', ie 'inertia' is not an INTRINSIC property of the charge as it depend both on μ0 and on A'! What this means is that it is a mistake to try to measure the 'mass' of a charged particle using a mass spectrometer, such as the one done by Kaufman!

2.)The current ,I, appearing in the equations is not necessarily the current in a good conductor such as a metal. In our case, it is a current in an insulator! It is simplistic to think of electric currents as only a feature of conductors, much like it is a mistake to think of motion to be just a feature of fluids inside a pipe. You will note that  though portions of a stiff object donnot flow like fluids, they still move slightly and then springs back. This is exactly how charges moves inside an insulator, even a perfect one, when they experience an electric field. They are pulled slightly away from the other charges before they are pulled back like in a spring, thereby though not being allwod to move throughout, like in a conductor, they are still allowed to vibrate or move slightly from their natural resting points. It is vibration that is all we need in wave propagation. The slight movement of charges inside insulators is still an electric current for the purposes of applying the Biot-Sarvat law and Faraday's Law.

To understand the EM wave, we need yet another equation that now indicates that the charges pulled apart can behave just like a spring when they 'fall back' on each other. To do this, begine by understanding the squared r appearing in 'inverse square law' as to be due to surface area of a sphere of radius r. So gravitational force, for instance, is, more acurately, stated in terms of mass per surface area, or 'mass density'. In our case, the 'law' in question is cuolomb's law, which can be stated as F=q^2/εA, where A is generally 'surface area across which the electrostatic force acts'. The case of inverse square law is a special case where A= 4πr^2 simply because the electrostatic force is spreading in all directions and thus crossing the entire area of some sphere of radius r. We don't use the standard inverse square law if the electrostatic forces (or 'field lines')are collimated as if passing through a pipe of area A. We can rewrite such 'cuolomb's law' as F=q^2x/εAx=(q^2/vol)x, where x is the distance, perpendicular to the surface of area A, moved by some charqe and thus vol=Ax is the volume scanned by the charge. So when you write coulumb's law in terms of 'charge density' (q/vol), it becomes just like the Hook's law for a spring. This shows that if a charge is being pulled away from another charge, so that there is a collimated electrostatic force between the two charges, then it behaves exactly like a spring when it gets pulled back by the electrostatic force. Here is how we relate the Cuolomb's Law with the Hook's law in a spring and thereby finds the ficticious 'spring's constant', k':

Note that:

1.In the first line, I introduce Cuolomb's law for the colimated electric field 'enclosed in a pipe or area A'). I also slightly rewrite it so that it is in terms of 'charge density',ρ, and by doing so, it immediately begins to look like Hook's Law, F=kx.

2.In the second line, I recognize the ficticious 'springs constant', k' by comparing the rewritten Cuolomb's Law with Hook's Law.

3.In the third line, I introduce the formular for the speed of waves, c, in a springy medium and then tried to rewrite the same equations by rather using the fictitious 'mass' and 'spring's constant' that we have seen. The result is that it shows that the speed of the waves in such a medium with ficticious 'spring constant' and 'inertia' is the speed of light!

So we can now combine this 'springiness' with the fictitious 'inertia' to figure out the kind of waves that can come about when this inertia and springiness interplay like we saw it the flaccid but springy ruler. This is Electromagetic waves! A neutral object is composed of charges of opposite signs that are sitting near each other. Whenever a force tries to displace the charges away from each other, the charges experiences a spring-like restoring force. When the charges are accelerating back to each other, they experiences a ficticious 'inertia'. The rest is understood like any other 'mechanical wave'! Also, this shows that 'empty space' is not empty at all, but is filled with an object closely similar to any other object, specifically, it is an insulator.